Rips filtrations for quasimetric spaces and asymmetric functions with stability results

Abstract

The Rips filtration over a finite metric space and its corresponding persistent homology are prominent methods in topological data analysis to summarise the “shape” of data. Crucial to their use is the stability result that says if $X$ and $Y$ are finite metric spaces then the (bottleneck) distance between the persistence diagrams constructed via the Rips filtration is bounded by $2d_{GH}\left(X,Y\right)$ (where $d_{GH}$ is the Gromov–Hausdorff distance). A generalisation of the Rips filtration to any symmetric function $f:X\times X\to ℝ$ was defined by Chazal, de Silva and Oudot (Geom. Dedicata 173 (2014) 193–214), where they showed it was stable with respect to the correspondence distortion distance. Allowing asymmetry, we consider four different persistence modules, definable for pairs $\left(X,f\right)$ where $f:X\times X\to ℝ$ is any real valued function. These generalise the persistent homology of the symmetric Rips filtration in different ways. The first method is through symmetrisation. For each $a\in \left[0,1\right]$ we can construct a symmetric function ${sym}_a\left(f\right)\left(x,y\right)=amin\left\{d\left(x,y\right),d\left(y,x\right)\right\}+\left(1-a\right)max\left\{d\left(x,y\right),d\left(y,x\right)\right\}$. We can then apply the standard theory for symmetric functions and get stability as a corollary. The second method is to construct a filtration $\left\{{\mathcal{ℛ}}^{dir}{\left(X\right)}_t\right\}$ of ordered tuple complexes where $\left(x_0,x_2,\dots ,x_p\right)\in {\mathcal{ℛ}}^{dir}{\left(X\right)}_t$ if $d\left(x_i,x_j\right)\le t$ for all $i\le j$. Both our first two methods have the same persistent homology as the standard Rips filtration when applied to a metric space, or more generally to a symmetric function. We then consider two constructions using an associated filtration of directed graphs or preorders. For each $t$ we can define a directed graph $\left\{D{\left(X\right)}_t\right\}$ where directed edges $x\to y$ are included in $D{\left(X\right)}_t$ whenever $max\left\{f\left(x,y\right),f\left(x,x\right),f\left(y,y\right)\right\}\le t$ (note this is when $d\left(x,y\right)\le t$ for $f=d$ a quasimetric). From this we construct a preorder where $x\le y$ if there is a path from $x$ to $y$ in $D{\left(X\right)}_t$. We build persistence modules using the strongly connected components of the graphs $D{\left(X\right)}_t$, which are also the equivalence classes of the associated preorders. We also consider persistence modules using a generalisation of poset topology to preorders.

The Gromov–Hausdorff distance, when expressed via correspondence distortions, can be naturally extended as a correspondence distortion distance to set–function pairs $\left(X,f\right)$. We prove that all these new constructions enjoy the same stability as persistence modules built via the original persistent homology for symmetric functions.